A class of strongly nonlinear functional differential equations
The existence of solutions of the functional differential equation \[ u'(t)+\partial \phi (u(t))\in F(t,u(t),u_ t),\quad 0<t<T,\quad u(s)=v(s),\quad -\tau \leq s\leq 0 \] is considered, where \(\partial \phi\) is the subdifferential of \(\phi\). Existence is proved by a fixed point argument based on the result that a function P: \(C\to C\), where C is a nonempty, convex and weakly compact subset of a separable locally convex space, which is weakly sequentially continuous has at least one fixed point. A theorem of Brézis shows that the equation \(u_ h'(t)=\partial \phi (u_ h(t))\ni h(t),\) \(u_ h(0)=v(0)\) has a solution for h in a certain ball in \(L^ 2\). Then the existence theorem follows by defining P by \((Ph)(t)=F(t,u_ h(t),\hat u_{h_ t}),\) and placing a certain demiclosedness condition on F. Here, \(\hat u(t)=v(t)\) for \(t\in [- \tau,0]\), \(\hat u(t)=u(t)\) for \(t\in (0,T_ 0]\), continuation of solutions is proved in the usual way and a number of examples are presented, including the Navier-Stokes equations with a delay in the viscosity.
scientific article; zbMATH DE number 4091956
scientific article